maps the non-autonomous Hietarinta equation (REF )
into equation (REF ).
The transformation (REF ) can be interpreted at
level of single cells as a simultaneous transformation of all
the variables associated to the vertices \(x\) , \(x_{1}\) , \(x_{2}\) 
and \(x_{12}\) , i.e., acting with an element of the group
\(\operatorname{M\ddot{o}b}^{4}\)  [1]}, [2]}, [3]}.
Following the explicit formula given in [3]}
the element of \(\operatorname{M\ddot{o}b}^{4}\)  corresponding to (REF )
is given by
\(x = -\frac{e_{1} X +e_{2}}{X+1},\quad x_{1} = -\frac{o_{1} X_{1} -e_{2}}{X_{1}-1},\quad x_{2} = -\frac{e_{1} X_{2}+o_{2}}{X_{2}+1},\quad x_{12} = -\frac{o_{1} X_{12}+o_{2}}{X_{12}-1}. \) 
